Download Geometry 1.6 Angle Pair Relationships

Geometry
1.6
Angle Pair Relationships
Essential Question: How can you use angle
pairs to find angle measures?
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Topics/Objectives
• Know what Vertical Angles are.
• Identify Linear Pairs.
• Solve problems with Complementary
Angles.
• Solve problems with Supplementary
Angles.
• To define adjacent angles.
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Geometry 1.6 Angle Pair Relationships
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Another way to name angles…
1
Sometimes, for
clarity and
convenience, we
will use a single
number inside
the angle to
name it.
This is 1.
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Geometry 1.6 Angle Pair Relationships
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More than one angle
1
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2
3
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Vertical Angles
1
3
2
4
Two angles are vertical
angles if their sides form
two pairs of opposite
rays.
1 & 2 are
vertical angles.
3 & 4 are
vertical angles.
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Vertical Angles Property
Vertical Angles are congruent.
60°
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?60°
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Linear Pair
Two adjacent angles Common Side
are a linear pair if
their noncommon
sides are opposite
1 & 2 are a linear pair.
rays.
2
1
Noncommon sides
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Linear Pair Property
The sum of the angles of a linear pair is 180°.
70°
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110°
?
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Example 1
2
1
5
3
4
a. Are 1 and 2 a linear pair? Yes
b. Are 4 and 5 a linear pair? no
c. Are 3 and 5 vertical angles? no
d. Are 1 and 3 vertical angles? Yes
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Example 2These angles form a
Find the measure of the
three angles.
linear pair. The sum
is 180°.
130°
2
50°
These are vertical
angles, and
congruent.
1 50°
3
130°
These angles are
vertical angles.
Vertical angles are
congruent.
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Example 3
A
B
(4x + 30)°
E
(6x – 10)°
D
page 1
Solve for x, then
find the measure of
each angle.
C
AEB and BEC form a linear pair.
What do we know about the sum of the angles of
a linear pair? The sum is 180°.
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Example 3
A
94°
B
(4x + 30)°
E
(6x – 10)°
86°
86°
94°
D
C
page 2
Linear pair AEB and BEC
means:
(4x + 30) + (6x – 10) = 180
10x + 20 = 180
10x = 160
x = 16
Then mAEB = 4(16) + 30 = 94
and mBEC = 6(16) – 10 = 86
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Your Turn
C
(5x + 30)° (2x – 4)°
A
B
145° 1
3
2
1. Find the measure
of 1, 2, 3.
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2. Find the measure
of ABC.
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Solutions
180° C
145° 1 35°
35° 3
2
145°
(5x + 30)° (2x – 4)°
A
B
5x + 30 + 2x – 4 = 180
7x + 26 = 180
7x = 154
x = 22
mABC = 5(22) + 30
= 140°
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Complementary Angles
Two angles are complementary if their
sum is 90°.
These angles are
complementary
and adjacent.
65°
25°
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Adjacent Angles
A
B
O
Adjacent angles have
the same vertex, O,
and one side in
common, OB. They
share no interior
points.
C
There are THREE angles:
You cannot use the
AOB or BOA
label O, since it
would be unclear
BOC or COB
which angle that is.
AOC or COA
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Geometry 1.4 Angles and Their Measure
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RST and VST are NOT
adjacent angles.
R
V
S
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T
Geometry 1.4 Angles and Their Measure
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Complementary Angles
Two angles are complementary if their sum is 90°.
These angles
are
complementary
and
30°
nonadjacent.
60°
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Supplementary Angles
Angles are supplementary if their sum is 180°.
The angles are
adjacent and
supplementary (and
a linear pair).
70°
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110°
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Supplementary Angles
Angles are supplementary if their sum is 180°.
The angles are
nonadjacent
and
supplementary.
80°
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100°
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Example
5(40) – 50 = 150°
(5y – 50)°
Solve for y, then
find m1.
30° 1
(4y – 10)°
150°
Vertical angles are congruent, so:
5y – 50 = 4y – 10
y =40
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1 forms a linear pair with
either of the 150° angles, so
1 is 30°.
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Example
Find the measure of each angle.
4x + 5 + 3x + 8 = 90
49°
(4x + 5)°
7x + 13 = 90
41°
(3x + 8)°
7x = 77
x = 11
This is a right angle,
the angles are
complementary. Their
sum is 90°.
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4(11) + 5 = 49°
3(11) + 8 = 41°
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Example 4
Find the value of each
50° (3x + 8)°
(5x – 20)° variable and the measure
50° of each labeled angle.
(5x + 4y)°
130°
3x + 8 = 5x – 20
130°
5x + 4y = 130
-2x = -28
5(14) + 4y = 130
x = 14
70 + 4y = 130
3(14) + 8 = 50°
4y = 60
y = 15
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Solve for x:
6 x  10  4 x  40
2 x  30
(4x + 40)
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(6x + 10)
x  15
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Solve for x:
(12x – 12)
(5x + 5)
(12 x  12)  (5 x  5)  180
17 x  7  180
17 x  187
x  11
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Solve for x:
(7x + 2)
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( x  8)  (7 x  2)  90
8 x  10  90
8 x  80
x  10
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Summarize
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